In the case that samples of independent path realizations of a SRH \(S\alpha S\)  process \(X\)  are available, consistent estimation of a symmetric spectral density \(f\)  of \(X\)  can be performed without any problems caused by the non-ergodicity of the process.
Consider the codifference function, which is the stable law analog of the covariance function when second moments do not exist.
It is defined by \(\tau (t)=2\Vert X(0)\Vert _\alpha ^\alpha -\Vert X(t)-X(0)\Vert _\alpha ^\alpha \) , where \(\Vert X(0)\Vert _\alpha \)  and \(\Vert X(t)-X(0)\Vert _\alpha \)  are the scale parameters of \(X(0)\)  and \(X(t)-X(0)\) , respectively, see [1]}.
Using the finite-dimensional characteristic function in Equation (REF ) these scale parameters can be computed explicitly which yields
\(\tau (t)=2\sigma ^\alpha - 2^\alpha \lambda _\alpha \int _{\mathbb {R}}\left|\sin \left(\frac{tx}{2}\right)\right|^\alpha m(dx),\) 
